A family of fourth-order parallel splitting methods for parabolic partial differential equations

Abstract

A family of numerical methods which are L-stable, fourth-order accurate in space and time, and do not require the use of complex arithmetic is developed for solving second-order linear parabolic partial differential equations. In these methods, second-order spatial derivatives are approximated by fourth-order finitedifference approximations, and the matrix exponential function is approximated by a rational approximation consisting of three parameters. Parallel algorithms are developed and tested on the one-dimensional heat equation, with constant coefficients, subject to homogeneous and time-dependent boundary conditions. These methods are also extended to two-and three-dimensional heat equations, with constant coefficients, subject to homogeneous boundary conditions.

Utilizing a third order difference scheme in numerical method of lines for solving the heat equation

“…Table 3 represents the numerical solution and absolute errors acquire by the proposed scheme at different values of z. Table 4 compares the computed errors of the current scheme to those of [3] for distinct time levels.…”

“…Here, h represents the step size in the z-direction, calculated as + Z M 1 , while k is the step size in the t-direction. The space derivative in (1) may be replace by five point third order finite difference approximation [3] for the mesh points (z, t) = (z m , t n ) with m = 1, 2, 3,...,M − 2. To achieve the same accuracy at the points (z M−1 , t n ) and (z M , t n ), special formulae are used which approximate…”

“…not only to the third order but also with dominant error term and for the point (z, t)=(z M , t n ), we have [3]:…”

“…Higher order numerical methods, which provide more accurate approximations of the underlying mathematical models, have emerged as a promising solution to address this demand. Additionally, the use of third order difference schemes [3] has been proven to effectively reduce numerical errors and enhance the overall accuracy of the numerical solution. In this study, we utilize MOL based on third order difference scheme for the solution of heat equation.…”

“…Discontinuities will exist if h(0) ≠ 0, or h(Z) ≠ 0. The solution to the problem (1)- (3) gives the temperature v at a distance of z units along a thin insulated bar after t units of time.…”